1 December 2008 Asymptotic stability of harmonic maps under the Schrödinger flow
Stephen Gustafson, Kyungkeun Kang, Tai-Peng Tsai
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Duke Math. J. 145(3): 537-583 (1 December 2008). DOI: 10.1215/00127094-2008-058


For Schrödinger maps from R2×R+ to the 2-sphere S2, it is not known if finite energy solutions can form singularities (blow up) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispersive sense, i.e., scatter) to a fixed harmonic map as time tends to infinity. The proof uses, among other things, a time-dependent splitting of the solution, the generalized Hasimoto transform, and Strichartz (dispersive) estimates for a certain two space--dimensional linear Schrödinger equation whose potential has critical power spatial singularity and decay. Along the way, we establish an energy-space local well-posedness result for which the existence time is determined by the length scale of a nearby harmonic map


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Stephen Gustafson. Kyungkeun Kang. Tai-Peng Tsai. "Asymptotic stability of harmonic maps under the Schrödinger flow." Duke Math. J. 145 (3) 537 - 583, 1 December 2008. https://doi.org/10.1215/00127094-2008-058


Published: 1 December 2008
First available in Project Euclid: 15 December 2008

zbMATH: 1170.35091
MathSciNet: MR2462113
Digital Object Identifier: 10.1215/00127094-2008-058

Primary: 35Q55
Secondary: 35B40

Rights: Copyright © 2008 Duke University Press


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Vol.145 • No. 3 • 1 December 2008
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